Let the topological space X be the so-called "long line" — which is an uncountable linearly ordered set containing all the countable ordinal numbers as well as a copy of the open unit interval (0,1) between each countable ordinal number and its successor. The topology of X is its order topology. Let the topological space Y be the 2-dimensional sphere-more precisely, let Y be the boundary of the closed unit ball in 3-dimensional Euclidean space. Let Z be the topological product of X and Y.
My questions are:
- Is Z 3-dimensional, and if not, what is its dimension?
- Does Z contain an uncountable collection of pairwise disjoint open subsets?
I mean by "dimension", the so-called "small inductive dimension". This is a well known function, which for any non-empty regular Hausdorff space (as argument) has a non-negative integer value (or is infinite). The function has the same value for any pair of homeomorphic spaces in its range of definition. This includes Z-which is regular and Hausdorff, since both X and Y are.
The motivation for my questions is this: The physical space of our universe is most simply and clearly pictured as a 3-dimensional Euclidean space E(3) extending to infinity in all directions-in other words, Newtonian space, before the complications of relativity theory arose. Now E(3) contains an uncountable infinity of points but can only contain a countable infinity of "disjoint discrete objects" such as stars, galaxies or even atoms.
What simply described topological space could serve as the space of a physical universe containing uncountably many "disjoint discrete objects"? Such a space should probably contain an uncountable set of pairwise disjoint open subsets. A non-separable Hilbert space would meet this condition but its dimension is infinite. Could there not exist a low dimensional space, closer to E(3), which also meets this condition? That is why I am asking these questions about the arc-wise connected space Z. I also wonder whether Z shares the following desirable property of Euclidean spaces-being homogeneous and isotropic. If p and q are distinct points of Z, does there always exist a homeomorphism of Z onto itself that takes p into q?